Ab initio description of the unbound 7 He

Transcription

1 Canada s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Ab initio description of the unbound 7 INT program INT--3 Light nuclei from first principles 5 th October 0, Institute for Nuclear Theory Petr Navratil TRIUMF Accelerating Science for Canada Un accélérateur de la démarche scientifique canadienne Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété d un consortium d universités canadiennes, géré en co-entreprise à partir d une contribution administrée par le Conseil national de recherches Canada

2 Outline Why 7? No-core shell model calculations for neutron rich isotopes Introducing no-core shell model with continuum (NCSMC 7 calculations: Comparison of NCSM/RGM and NCSMC 7 predictions and comparison to experiment Outlook

3 Unbound exotic 7 Experimental situation 3/ - g.s. resonance at 0.43 MeV above n + Borromean halo system 5/ - resonance established Controversy about / - resonance Low-lying narrow Broad at 3 MeV Extremely broad Experiments very challenging: three-body background 3

4 Unbound exotic 7 Experimental situation Controversy about / - resonance Low-lying narrow [ 8 + C fragmentation] Broad at 3 MeV [d(,p 7, H( 8 Li, 3 7 ] Extremely broad [p+ : isospin analog] J π experiment E R Γ Ref. 3/ 0.430(3 0.8(5 [] 5/ 3.35(0.99(7 [40] / 3.03(0 [] [5].0( 0.75(8 [5] [5] M. Meister et al., Phys.Rev.Lett.88, 050(00. [] A. H. Wuosmaa et al., Phys.Rev.C7, 030(005. [4] Yu. Aksyutina, Phys. Lett. B, 9(009. [5] P. Boutachkov et al., Phys. Rev. Lett. 95, 350(005. Ab initio calculations based on bound-state techniques cannot give any insight 4

5 Chiral Effective Field Theory First principles for Nuclear Physics: QCD Non-perturbative at low energies Lattice QCD in the future For now a good place to start: Inter-nucleon forces from chiral effective field theory Based on the symmetries of QCD Chiral symmetry of QCD (m u m d 0, spontaneously broken with pion as the Goldstone boson Degrees of freedom: nucleons + pions Systematic low-momentum expansion to a given order (Q/Λ χ Hierarchy Consistency Low energy constants (LEC Fitted to data Can be calculated by lattice QCD Λ χ ~ GeV : Chiral symmetry breaking scale

6 The ab initio no-core shell model (NCSM The NCSM is a technique for the solution of the A-nucleon bound-state problem Realistic nuclear Hamiltonian High-precision nucleon-nucleon potentials Three-nucleon interactions N = N max + Finite harmonic oscillator (HO basis A-nucleon HO basis states complete N max hω model space Effective interaction tailored to model-space truncation for NN(+NNN potentials Okubo-Lee-Suzuki unitary transformation Or a sequence of unitary transformations in momentum space: Similarity-Renormalization-Group (SRG evolved NN(+NNN potential A Ψ A = N max N=0 i A c Ni Φ Ni Convergence to exact solution with increasing N max for bound states. No coupling to continuum.

7 4 from chiral EFT interactions: g.s. energy convergence 4 E [MeV] bare (3 SRG (.0/8 N 3 LO (500 MeV NN + NNN Chiral N 3 LO NN plus N LO NNN potential Bare interaction (black line Strong short-range correlations Large basis needed SRG evolved effective interaction (red line Unitary transformation 8 ( H α =U α H U + α dh α dα = "# [ T, H α ], H α $ % α = λ N max A=3 binding energy and half life constraint c D =-0., c E =-0.05, Λ=500 MeV Two- plus three-body components, four-body omitted Softens the interaction Smaller basis sufficient

8 4 8 and 8 with SRG-evolved chiral N 3 LO NN + N LO NNN 3N matrix elements in coupled-j single-particle basis: Introduced and implemented by Robert Roth et al. A=3 binding energy & half life constraint c D =-0., c E =-0.05, Λ=500 MeV Now also in my codes: Jacobi-Slater-Determinant transformation & NCSD code Example:, 8 NCSM calculations up to N max =0 done with moderate resources E gs [MeV] NN+3N(500 SRG Λ=.7 fm - hω= MeV NCSM Exp fit extrap N max E gs [MeV] NN+3N(500 SRG Λ=.7 fm - hω= MeV NCSM Exp fit extrap N max 8

9 4 8 and 8 with SRG-evolved chiral N 3 LO NN + N LO 3N chiral N 3 LO NN: 4 underbound, and 8 unbound chiral N 3 LO NN + N LO 3N(500: 4 OK, both and 8 bound -4 A=3 binding energy & half life constraint c D =-0., c E =-0.05, Λ=500 MeV NNN interaction important to bind neutron rich nuclei E gs [MeV] NN NN+3N(500 Expt A 9

10 and 8 with SRG-evolved chiral N 3 LO NN + N LO 3N chiral N 3 LO NN: 4 underbound, and 8 unbound chiral N 3 LO NN + N LO 3N(400: 4 fitted, barely unbound, 8 unbound describes quite well binding energies of C, O, 40 Ca, 48 Ca chiral N 3 LO NN + N LO 3N(500: 4 OK, both and 8 bound does well up to A=0, overbinds C, O, Ca isotopes SRG-N 3 LO NN Λ=.0 fm - : 4 OK, both and 8 bound O, Ca strongly overbound 4 binding energy & 3 H half life constraint c D =-0., c E =+0.098, Λ=400 MeV A=3 binding energy & half life constraint c D =-0., c E =-0.05, Λ=500 MeV NNN interaction important to bind neutron rich nuclei E gs [MeV] NN NN+3N(400 NN+3N(500 NN-srg.0 Expt 4 8 Our knowledge of the 3N interaction is incomplete A 0

11 NCSM calculations of and 7 g.s. energies - -0 E gs [MeV] SRG-N 3 LO NN Λ=.0 fm - N max = N max = 4 N max = N max = 8 N max =0 N max = extrap hω [MeV] E gs [MeV] NCSM Exp fit extrap SRG-N 3 LO NN Λ=.0 fm - hω= MeV 7 3/ N max 0 + ü N max convergence OK ü Extrapolation feasible E g.s. [MeV] : E gs =-9.5(5 MeV (Expt MeV 7 : E gs =-8.7(5 MeV (Expt (30 MeV 7 unbound (+0.430(3 MeV, width 0.8(5 MeV NCSM: no information about the width 4 7 NCSM N max = NCSM extrap. -8.( -9.5(5-8.7(5 Expt unbound

12 The ab initio NCSM/RGM in a snapshot Ansatz: eigenstates of H (A-a and H (a in the ab initio NCSM basis Many-body Schrödinger equation: ê realistic nuclear Hamiltonian Hamiltonian kernel Norm kernel

13 Consider the T = ½ case: 5 ( 5 Li Five-nucleon cluster unbound; 4 tightly bound, not easy to deform 4 n H n d + p Energy 3 H MeV Satisfactory description of n- 4 ( p- 4 scattering at low excitation energies within single-channel approximation However, both n(p + 4 and d + 3 H( 3 channels needed to describe 3 H(d,n 4 [ 3 (d,p 4 ] fusion!

14 Unbound A=5 nuclei: 5 è n+ 4, 5 Liè p+ 4 NCSM/RGM calculations 4 SRG-N 3 LO NN potential with Λ=.0 fm - n A y (! c.m E n = 7 MeV d"/d#(! c.m NCSM/RGM Krupp et al (n,n 4 Differential cross section and analyzing MeV neutron energy Polarized neutron experiment at Karlsruhe 50 E n = 7 MeV ! c.m. NNN missing: Good agreement only for energies beyond low-lying 3/ - resonance 4

15 How about 7 as n+? SRG-N 3 LO Λ =.0 fm h- Ω = MeV E x [MeV] ( +, -, h - Ω 4h - Ω h - Ω 8h - Ω 0h - Ω h - Ω Expt All excited states above + broad resonances or states in continuum Convergence of the NCSM/RGM n+ calculation slow with number of states Negative parity states also relevant Technically not feasible to include more than ~ 5 states 5

16 New approach: NCSM with continuum NCSM/ Ψ A J π T = c Ni ANiJ π T Ni

17 New developments: NCSM with continuum NCSM/ NCSM/RGM Ψ A J π T = c Ni ANiJ π T Ni The idea behind the NCSMC r J T A = X Z d r ( râ J T (A a,a r H = EN = N + (N HN = E NCSMC HNCSM + J T A = X c A J T + X h r Z h N HN d r X Z 0 c d r 0 N 0 ( r, r 0 0( r 0 ḡ = E ḡ! Â J T (A r c a,a

18 New developments: NCSM with continuum NCSM/ NCSM/RGM Ψ A J π T = c Ni ANiJ π T Ni The idea behind the NCSMC r J T A = X Z d r ( râ J T (A a,a r H = EN = N + (N HN = E NCSMC HNCSM + J T A = X c A J T + X h r Z h N HN d r X Z 0 c d r 0 N 0 ( r, r 0 0( r 0 ḡ = E ḡ! Â J T (A r c a,a

19 NCSMC formalism Start from ( HNCSM h h H ( c χ = E ( ḡ ḡ ( c χ NCSM sector: (H NCSM λλ = AλJ π T Ĥ Aλ J π T = ε J π T λ δ λλ NCSM/RGM sector: H νν (r, r = µµ dydy y y N νµ (r, yh µµ (y, y N µ ν (y,(,r 9

20 ψ J π T = ν g ν J πt (r r ( (st Y ( ˆr A a,a π Â ν A a α I π T a α I T (J π T δ(r r A a,a rr A a,a r dr J Φ π T νr (Jacobi channel basis Since we are using NCSM wave functions, it is convenient to introduce Jacobi channel states in the HO space J Φ π T νn ( (st Y ( ˆr A a,a π = A a α I π T a α I T (J π T R n (r A a,a The coordinate space channel states are given by J Φ π T νr = R n (r We used the closure properties of HO radial wave functions n J Φ π T νn δ(r r A a,a = R n (rr n (r A a,a n Target and projectile wave functions are both translational invariant NCSM eigenstates calculated in the Jacobi coordinate basis

21 J Φ π T ν r  J π T v  v Φ νr = ( A r ( a = ( a = ( A r J N πt δ( r r v v ( r,r = δ v v r r (A R n ( r R n (r Φ n n J π T ν n J ˆPA,A Φ π T νn Direct term: Treated exactly! (in the full space ν ν ( A ( A ( a = SD ψ µ (A a + aψ ν Exchange term: Obtained in the model space! (Many-body correction due to the exchange part of the intercluster antisymmetrizer δ(r r A a,a = R n (rr n (r A a,a n (A SD

22 J Φ π T ν r  J π T v HÂv Φ νr ( A A = H ˆPiA r ( a = i= ( a = ( A r J H πt v v ( r,r = T rel (r+v Coul (r+ε α π I T N J πt v v ( r,r + (A R n ( r R n (r Φ n n J πt ν n V A,A (A (A R n ( r R n (r Φ n n J π T ν n ( ˆP J A,A Φ π T νn J ˆPA,A V A,A Φ π T νn + ( A ( A (A - Direct potential: in the model space (interaction is localized! Exchange potential: in the model space

23 Define SD channel states in which the eigenstates of the heaviest of the two clusters (target are described by a SD wave function: J Φ π T νn SD ( (st (a Y ˆRc.m. π = A a α I π T a α SD I T ( (J π T R n (a ( R c.m. (A a R c.m. O π A a α I (A a T ϕ 00 ( Rc.m. Vector proportional to the c.m. coordinate of the a nucleons Vector proportional to the c.m. coordinate of the A-a nucleons (a R c.m. R (A c.m. ξ 0 (A a (a ( ϕ 00 ( Rc.m. ϕ n ( Rc.m. = 00, n, n r r, NL, d= a n r r,nl O A a η A a ϕ nr ηa a r ( ϕ NL ξ0 η = ( (A aa r A a A A a,a ( (

24 More in detail: J Φ π T νn SD s = ˆ Ĵ r ( s+ r+l+j r J r L J n r r,nl,j r 00, n, n, NL, r r d= a The spurious motion of the c.m. is mixed with the intrinsic motion A a π J rt r ϕ NL ( ξ 0 ( J π T Φ νr n r SD J Φ π T ν n Calculate these Matrix that can be inverted J Ô Φ π T νn SD = Φ n r r,n r r,j r J r π r T νr n r ˆ ˆ NL π J r Ô Φ r T νr n r Ĵr ( s+ s s r J r L J Interested in this Transformation is general: same for different A s or different a s s r J r L J 00, n, n r r, NL, d= a d = A a A a a Translational invariance preserved (exactly! also with SD channels 00, n, n r r, NL,

25 SD to Jacobi transformation is general and exact Can use powerful second quantization representation Matrix elements of translational invariant operators can be expressed in terms of matrix elements of density operators on the target eigenstates For example, for a =a = SD Φ ν n One-body density matrix elements J πt P A,A Φ νn J π T SD I s J j = A j j Kτ ŝ ŝ ĵ ĵ I s J j ˆK ˆ τ ( I + j +J ( T + +T I K I j J j T τ T T SD A α π I ( (Kτ π n j A α I T SD + T a n j a

26 Start from ( ( HNCSM h c = E h H χ NCSMC formalism ( ( ḡ ḡ ( c χ Coupling: ḡ λν (r = ν dr r AλJ π T Âν Φ J π T ν r N ν ν (r,r h λν (r = ν dr r AλJ π T ĤÂν ΦJ π T ν r N ν ν (r,r. Calculation of g from SD wave functions: g λνn = AλJ π T ÂνΦ J π T νn = nl00, l 00nl, l (A j ( I +J+jŝĵ { I s l J j } Ĵ ˆT AλJ π T a nlj Φ J π T νn SD

27 7 spectroscopic factors Obtained as S λν = n g λνn Not the final result to be compared to experiment, rather input in the NCSMC calculations 7 J π n(lj NCSM CK VMC GFMC Exp. 3/ 0 + p (8[3] 0.4(9 [50] 0.37(7 [45] 3/ + p / + p / + p / + p / 0 + p / + p / + p / + p / + p / + p 0. 5/ + p / 0 + p / + p / + p / + p / + p 3 0.5

28 NCSMC formalism Start from ( HNCSM h h H ( c χ = E ( ḡ ḡ ( c χ ( ( ( νrν r = δλλ ( ḡ λν (r ḡ λ ν(r δ νν δ(r r N λλ rr ( Orthogonalization: H = N ( HNCSM h h H N ( c χ = N + ( c χ Solve with generalized microscopic R-matrix Bloch operator ( ( (Ĥ + ˆL E = c χ ˆL c χ ˆL ν = ( δ(r a( d dr B ν r 8

29 Separation into internal and external regions at the channel radius a Internal region External region u c (r = A cn f n (r u c (r ~ v c δ ci I c (k c r U ci O c (k c r n [ ] 0 a r r c This is achieved through the Bloch operator: System of Bloch-Schrödinger equations: L c = d δ(r a µ c dr B c r T ˆ [ rel (r + L c + V Coul (r (E E c ]u c (r + dr'r'w cc' (r,r'u c' (r' Internal region: expansion on square-integrable basis External region: asymptotic form for large r c' u c (r = = L c u c (r n A cn f n (r [ ] u c (r ~ C c W (k c r or u c (r ~ v c δ ci I c (k c r U ci O c (k c r Scattering matrix Bound state Scattering state

30 After projection on the basis f n (r: [ C cn,c'n' (E E c δ cn,c'n' ]A c'n' = k c f / n L c I c δ ci U ci O c µ c v c c'n' f n ˆ T rel (r + L c + V Coul (r f n' δ cc' + f n W cc' (r,r' f n'. Solve for A cn. Match internal and external solutions at channel radius, a [ c aδ ci U c i O c (k c a ] = c k R c a c c I c (k µ c v c µ c v c I c (k c aδ ci U ci O c (k c a [ ] Lagrange basis associated with Lagrange mesh: { ax n [0,a]} g(xdx λ n g(x n In the process introduce R-matrix, projection of the Green s function operator on the channel-surface functions 0 0 a N n= f n (r f n' (rdr δ nn' R cc' = nn' µ c a f n (a[ C EI] cn,c'n' µ c' a f n' (a

31 3. Solve equation with respect to the scattering matrix U [ c aδ ci U c i O c (k c a ] = c k R c a c c I c (k µ c v c µ c v c I c (k c aδ ci U ci O c (k c a [ ] 4. You can demonstrate that the solution is given by: U = Z Z *, Z c c = ( k c a O c (k c aδ c c k c a R c c O c (k c a Scattering phase shifts are extracted from the scattering matrix elements U = exp(iδ δ

32 NCSM with continuum: 7 +n δ [deg] P3/ δ [deg] P3/ n n E kin [MeV] E kin [MeV] NCSM/RGM with up to three states NCSMC with up to three states and four 7 eigenstates More 7-nucleon correlations Fewer target states needed Expt. 3

33 NCSM with continuum: 7 +n 50 0 (a (b P/ δ [deg] P/ n+ δ [deg] n E kin [MeV] E kin [MeV] NCSM/RGM with up to three states NCSMC with up to three states and three 7 eigenstates More 7-nucleon correlations Fewer target states needed Expt. 33

34 NCSM with continuum: 7 +n 50 0 P3/ P5/ (a 50 0 P3/ P5/ (b δ [deg] 0 30 P/ P3/ δ [deg] 0 30 P/ P3/ 0-30 S/ n+ (g.s S/ n+ (g.s E kin [MeV] E kin [MeV] NCSM/RGM with three states NCSMC with three states and ten 7 eigenstates More 7-nucleon correlations Fewer target states needed Expt. 34

35 7 : NCSMC vs. NCSM/RGM vs. NCSM E [MeV] ? 7 SRG-N 3 LO Λ =.0 fm 5/ - / - 3/ - +n Expt NCSMC NCSM/RGM NCSM 3/ - 5/ - / - 3/ - J π experiment NCSMC NCSM/RGM NCSM E R Γ Ref. E R Γ E R Γ E R 3/ 0.430(3 0.8(5 [] / 3.35(0.99(7 [40] / 3.03(0 [] [5].0( 0.75(8 [5] [] A. H. Wuosmaa et al., Phys.Rev.C7, 030(005. NCSMC and NSCM/RGM energies where phase shift derivative maximal NCSMC and NSCM/RGM widths from the derivatives of phase shifts Γ = δ(e kin / E kin Ekin =E R Experimental controversy: Existence of low-lying / - state not seen in these calculations Best agreement with the neutron pick-up and proton-removal reactions experiments [] 35

36 Predictions of other resonances Two 3/ - resonances predicted at about 3.7 MeV and.5 MeV with widths of.8 MeV and 4.3 MeV, respectively Experiment: State of undetermined spin and parity at.(3 MeV with the width of 4( MeV Considerable mixing of P-waves in 3/ P3/ P5/ P3/ 0 P3/ 0 δ [deg] 30 P3/ δ [deg] P5/ n+ (g.s P3/ n+ (g.s P3/ E kin [MeV] E kin [MeV] NCSMC eigenphase shifts NCSMC diagonal phase shifts 3

37 Conclusions and Outlook We developed a new unified approach to nuclear bound and unbound states Merging of the NSM and the NCSM/RGM We demonstrated its capabilities in calculations of 7 resonances We find reasonable agreement with experiment for established 3/ - and 5/ - resonances Our results do not support the existence of a low lying narrow /- resonance We predict two broad 3/ - resonances arxiv: Outlook: Inclusion of 3N interactions Extension of the formalism to composite projectiles (deuteron, 3 H, 3, 4 Extension of the formalism to coupling of three-body clusters

38 NCSMC and NCSM/RGM collaborators S. Baroni (ULB Sofia Quaglioni (LLNL Robert Roth, Joachim Langhammer (TU Darmstadt C. Romero-Redondo, F. Raimondi (TRIUMF G. Hupin, M. Kruse (LLNL W. Horiuchi (Hokkaido